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\newcommand{\E}{{\tt E}}

\title[Harvard EconCS Seminar]{Budget Feasible Mechanism\\for Experimental
Design}
\author[Thibaut Horel]{Thibaut Horel, Stratis Ioannidis, and S. Muthukrishnan}

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\newcommand{\ie}{\emph{i.e.}}
\newcommand{\eg}{\emph{e.g.}}
\newcommand{\etc}{\emph{etc.}}
\newcommand{\etal}{\emph{et al.}}
\newcommand{\reals}{\ensuremath{\mathbb{R}}}

\begin{document}

\maketitle

\section{Motivation}

\begin{frame}{Motivation}
   \begin{center}
        \includegraphics<1>[scale=0.4]{stg-8.pdf}
        \includegraphics<2>[scale=0.4]{stg-7.pdf}
        \includegraphics<3>[scale=0.4]{stg-6.pdf}
        \includegraphics<4>[scale=0.4]{stg-5.pdf}
        \includegraphics<5>[scale=0.4]{stg-4.pdf}
        \includegraphics<6>[scale=0.4]{stg-3.pdf}
        \includegraphics<7>[scale=0.4]{stg-2.pdf}
        \includegraphics<8>[scale=0.4]{stg-1.pdf}
   \end{center}

\end{frame} 

\begin{frame}{Application and Challenges}
    \begin{itemize}
	\item<1-> Applications
            \begin{itemize}
		\item Medicine/Sociology
		\item Online surveys
		\item Data markets
	    \end{itemize}
\vspace*{1cm}
        \item<2-> Challenges
	     \begin{itemize}
	     	\item Which users are \alert{the most valuable}?
            \item What if users are \alert{strategic}?
        \end{itemize}
    \end{itemize}
\end{frame}

\begin{frame}{Answers}
    \begin{itemize}
        \item Which users are \alert{the most valuable}?
            \pause
            \begin{itemize}
                \item Experimental Design
            \end{itemize}
            \pause
        \item Experimental design with \alert{strategic agents}?\pause
            \begin{itemize}
                \item Budget Feasible Mechanisms [Singer, 2010]
            \end{itemize}
    \end{itemize}
	\pause
	\vspace*{1cm}

    For Linear Regression:
	    \begin{itemize}
            \item We present a deterministic, poly-time, truthful, budget
                feasible, 12.9-aproximate mechanism. 
            \item Lower bound of 2.
	    \end{itemize}
\end{frame}

\section{Experimental design}

\begin{frame}{Outline}
\tableofcontents
\end{frame}

\begin{frame}{Outline}
    \tableofcontents[currentsection]
\end{frame}

\begin{frame}{Problem}
   \begin{center}
        \includegraphics<1>[scale=0.4]{st2.pdf}
        \includegraphics<2>[scale=0.4]{st3.pdf}
        \includegraphics<3>[scale=0.4]{st4.pdf}
        \includegraphics<4>[scale=0.4]{st5.pdf}
        \includegraphics<5>[scale=0.4]{st6.pdf}
        \includegraphics<6>[scale=0.4]{st6a.pdf}
        \includegraphics<7>[scale=0.4]{st6b.pdf}
        \includegraphics<8>[scale=0.4]{st6c.pdf}
        \includegraphics<9>[scale=0.4]{st6d.pdf}
        \includegraphics<10->[scale=0.4]{st6e.pdf}
    \end{center}
	\vspace*{-2em}
    \begin{center}
        \begin{overprint}
        \onslide<1>\begin{center}$N$ ``experiment subjects'', $[N]\equiv
        \{1,\ldots,N\}$ \end{center}
            \onslide<2>
            \begin{center}
                $x_i\in \reals^d$: public features (\eg, age, gender, height, \etc)
            \end{center}
            \onslide<3>
            \begin{center}
                $y_i\in \reals$: private data (\eg, survey answer, medical test outcome, movie rating\ldots)
            \end{center}
            \onslide<4>
                \textbf{Gaussian Linear Model.}  There exists $\beta\in \reals^d$ s.t.\vspace*{-1em}
    \begin{displaymath}
        y_i = \beta^T x_i + \varepsilon_i,\quad
        \varepsilon_i\sim\mathcal{N}(0,\sigma^2), \quad i\in [N]
    \end{displaymath}
	    \onslide<5>
	    \begin{center}
		Experimenter {\tt E} wishes to learn $\beta$. 
	    \end{center}
       \onslide<6>
       \begin{center}
	Each subject $i\in [N]$ has a cost $c_i\in \reals_+$
	\end{center}
            \onslide<7>
             \begin{center}
                \E\ has a budget $B$.
            \end{center}
	    \onslide<8-9>
	    \begin{center}
		\E\ pays subjects\visible<9>{; $y_i$ is revealed upon payment.}
	    \end{center} 
	    \onslide<10>
	    \begin{center}
            {\tt E}  estimates ${\beta}$  through \emph{ridge regression}.
	    \end{center}
	    \onslide<11>
	    \begin{center}
            Goal: Determine who to pay how much so that $\hat{\beta}$ is as
            accurate as possible.
        \end{center}
        \end{overprint}
    \end{center}
\end{frame}

\begin{frame}{Which Users Are Informative?}

Let $S\subseteq [N]$ be the set of selected users.

\pause

\E\ has a \emph{prior} on $\beta$:
$$\beta \sim \mathcal{N}(0,\sigma^2 R^{-1}). $$

\pause

MAP estimation after observing private values:
\begin{align*}
    \hat{\beta} = \argmax_{\beta\in\reals^d} \mathbf{P}(\beta\mid y_i, i\in S)
    \pause
=\argmin_{\beta\in\reals^d} \big(\sum_{i\in S} (y_i - {\beta}^Tx_i)^2
    + \beta^TR\beta\big)
\end{align*}

\pause

Variance:
\begin{displaymath}
    \mathrm{Var}\, \hat{\beta} = \Big(R+\sum_{i\in S}x_ix_i^T\Big)^{-1} 
\end{displaymath}

\pause

Minimizing variance: which scalarization?\pause{} \alert{D}eterminant (\alert{D}-optimal design)
\begin{displaymath}
    \det\Big(R+\sum_{i\in S}x_i x_i^T\Big)^{-1}
\end{displaymath}
\end{frame}

\begin{frame}{Which Users Are Informative? (contd.)}
Let $S\subset [N]$ be the set of selected users.

\E\ has a \emph{prior} on $\beta$:
$$\beta \sim \mathcal{N}(0,\sigma^2 R^{-1}). $$

\pause
Information Gain: reduction of uncertainty on $\beta$
\begin{align*}
    I(\beta;S)&=H(\beta)-H(\beta\mid y_i,i\in S)\\
    \pause
    & = \frac{1}{2}\log\det \Big(R+\sum_{i\in S}x_ix_i^T\Big)-\frac{1}{2}\log\det R
\end{align*}

\pause
\begin{alertblock}{}
    \begin{center}
Maximizing information = Minimizing variance
\end{center}
\end{alertblock}
\end{frame}

\begin{frame}{Value function}
\begin{align*}
V(S)=\log\det\Big(R+\sum_{i\in S}x_ix_i^T\Big)
\end{align*}
\pause

Marginal contribution of a user:
\begin{align*}
    &V(S\cup\{i\})-V(S) = \log(1+x_i^TA_S^{-1}x_i),\\
    &\text{where } A_S = R + \sum_{i\in S}x_ix_i^T
\end{align*}
\pause
\begin{itemize}
\item Increase is greatest when $x_i$ \alert{spans a new direction}
    \pause
\item Adding an experiment \alert{always helps}
    \pause
\item $V$ is \alert{submodular}:
$$V(S\cup\{i\})-V(S) \geq  V(S'\cup\{i\})-V(S'),\quad \text{ for }S\subseteq S'. $$
\end{itemize}
\end{frame}


\begin{frame}{Experimental Design}
\begin{block}{\textsc{Experimental Design Problem (EDP)} }
\vspace*{-0.4cm}		
\begin{align*}
			\text{Maximize}\qquad &V(S) = \log \det (\alert<2>{I}+\sum_{i\in S}x_ix_i^T) \\
			\text{subj. to}\qquad &\textstyle\sum_{i\in S}c_i\leq B
\end{align*}
\end{block}
\pause
\begin{itemize}
\item $R=I$: homotropic prior
    \pause
\item EDP is NP-hard 
    \pause
\item $V$ is submodular, monotone, non-negative, and $V(\emptyset)=0$
    \pause
\item $\frac{e}{e-1}$-approximable (Sviridenko 2004, Krause and Guestrin 2005)
\end{itemize}
\end{frame}

\begin{frame}{Constant-approximation algorithm}
    \begin{block}{Budgeted Submodular Maximization}
        \begin{itemize}
        \item Let $i^* = \arg\max_i V(\{i\})$
        \item Compute $S_G$ greedily:
            \begin{itemize}
                \item repeatedly add element with \emph{highest
                    marginal-value-per-cost}:
                    \begin{align*}
                        i = \argmax_{j\in[N]\setminus
                        S}\frac{V(S\cup\{i\}) - V(S)}{c_i}
                    \end{align*}
                \item stop when the budget is exhausted
            \end{itemize}
        \item Return:
		\begin{itemize}
		\item $i^*$ if $V(\{i^*\})>V(S_G)$
		\item $S_G$ otherwise
		\end{itemize}
	\end{itemize}
    \end{block}
\vspace*{2em}
\pause

This algorithm is:
\begin{itemize}
\item poly-time,
    \pause
\item $\frac{5e}{e-1}$-approximate.
\end{itemize}
\end{frame}

\begin{frame}{Summary}
\begin{block}{\textsc{Experimental Design Problem (EDP)} }
\vspace*{-0.4cm}		
\begin{align*}
			\text{Maximize}\qquad &V(S) = \log \det (I+\sum_{i\in S}x_ix_i^T) \\
			\text{subj. to}\qquad &\textstyle\sum_{i\in S}c_i\leq B
\end{align*}
\end{block}
\vspace{1cm}
\pause
Simple $\frac{2e}{e-1}$-approximate algorithm.
\end{frame}
\section{Experimental design with strategic users}

\begin{frame}{Outline}
    \tableofcontents[currentsection]
\end{frame}

\begin{frame}{Strategic users}
\begin{center}
\includegraphics<1->[scale=0.3]{st10c.pdf}
\end{center}
\begin{itemize}
\visible<2-3>{\item
Subjects are \alert{strategic} and may lie about costs $c_i$.}
\visible<3>{\item
Subjects \alert{do not lie} about $y_i$ (tamper-proof experiments).}
\visible<4-5>{\item Goal: estimate $\beta$ accurately.}
\visible<5>{\item Adapt selection algorithm and payments.}
\end{itemize}
\end{frame}

\begin{frame}{Budget Feasible Mechanism Design [Singer 2010]}
Let $S\subset [N]$ be the selected users, and
  $c=[c_i]_{i\in [N]}$ be the reported costs.  \\\bigskip
\visible<2->{Let $V(S)\in\reals_+$ denote the \alert{value} of the experiments.}
\visible<3->{$$\text{High }V(S) \Leftrightarrow \text{ estimate }\hat{\beta}(S) \text{ is good}$$}
\visible<4->{
\begin{block}{Reverse Auction Mechanism}
 A mechanism $\mathcal{M}(c)=(S(c),p(c))$ comprises
\begin{itemize}
\item an \alert{allocation function} $S:\reals_+^N\to 2^{[N]}$, and
\item a \alert{payment function} $p:\reals_+^N\to \reals_+^N$.
\end{itemize}
\end{block}
}
\end{frame}


\begin{frame}{Budget Feasible Mechanism Design [Singer 2010] }
    We seek mechanisms $\mathcal{M}=(S,p)$  that are: 
    \pause
    \vspace{0.3cm}
    \begin{itemize}
	\item Normalized: $p_i=0$ if $i\notin S$.
            \vspace{0.2cm}
            \pause
        \item Individually Rational: $p_i\geq c_i,\;i\in S$
            \vspace{0.2cm}
            \pause
        \item Truthful
            \vspace{0.2cm}
            \pause
        \item budget feasible: $\sum_{i\in S} p_i \leq B$
            \vspace{0.2cm}
    \end{itemize}

    \pause
    \vspace{0.3cm}

    In addition, $\mathcal{M}$ must be:
    \vspace{0.3cm}
    \pause
    \begin{itemize}
        \item computationally efficient: polynomial time
            \pause
            \vspace{0.2cm}
        \item approximation: $OPT \leq \alpha V(S)$  with:
            \begin{displaymath}
                OPT = \max_{S\subset \mathcal{A}} \left\{V(S)\mid \sum_{i\in S}c_i\leq B\right\}
            \end{displaymath}
    \end{itemize}
\end{frame}

\begin{frame}{Strategic Subjects}
    When $V$ is submodular:
 %   \vspace{1cm}
    \pause
    \begin{itemize}
        \item \alert{Randomized}, poly-time, universally truthful  mechanism,
            approximation ratio: $7.91$ [Singer 2010, Chen \emph{et al.}, 2011]
        \vspace{0.5cm}
        \pause
        \item Deterministic, \alert{exponential time}, truthful  mechanism,
            approximation ratio: $8.34$ [Singer 2010, Chen \emph{et al.}, 2011]
        \vspace{0.5cm}
        \pause
        \item \alert{Deterministic, poly-time}, truthful mechanisms for specific submodular functions $V$ :
            \vspace{0.3cm}
            \begin{itemize}
                \item \textsc{Knapsack}: $2+\sqrt{2}$ [Singer 2010, Chen \emph{et al.}, 2011]
                    \vspace{0.3cm}
                \item \textsc{Matching}: 7.37 [Singer, 2010]
                    \vspace{0.3cm}
                \item \textsc{Coverage}: 31 [Singer, 2012]
                    \vspace{0.3cm}
            \end{itemize}
    \end{itemize}
\end{frame}

\section{Main Results}

\begin{frame}{Outline}
    \tableofcontents[currentsection]
\end{frame}


\begin{frame}{Our Main Results}
   \begin{theorem}<2->
        There exists a deterministic,  budget feasible, individually rational and truthful mechanism for
        EDP which runs in polynomial time. Its
        approximation ratio is:
        \begin{displaymath}
        \frac{10e-3 + \sqrt{64e^2-24e + 9}}{2(e-1)}
        \simeq 12.98
        \end{displaymath}
    \end{theorem}
   \begin{theorem}<3->
        There is no 2-approximate, budget feasible, individually rational and truthful mechanism for
        EDP.     \end{theorem}

\end{frame}

\begin{frame}{Myerson's Theorem}
\begin{theorem}[Myerson 1981]
 A normalized mechanism $\mathcal{M} = (S,p)$ for a single parameter auction is
truthful iff:
\begin{enumerate}
\item<2-> 
 $S$ is \alert{monotone}, \emph{i.e.},%\\ for any agent $i$ and $c_i' \leq c_i$, for any fixed costs $c_{-i}$ of agents in $[N]\setminus\{i\}$,
 $$\text{for all}~c_i' \leq c_i,\quad i\in S(c_i,
c_{-i})\text{ implies }i\in S(c_i', c_{-i}),$$ and 
\item<3->
 subjects are paid \alert{threshold payments}, \emph{i.e.}, $$\text{for all }i\in S(c), \qquad p_i(c)=\inf\{c_i': i\in S(c_i', c_{-i})\}.$$
\end{enumerate}
\end{theorem}
\uncover<4->{Focus on  \alert{monotone} mechanisms.}
\end{frame}

\begin{frame}{Can we use submodular maximization?}
    \begin{block}{Allocation rule candidate}
        \begin{itemize}
        \item Compute $i^* = \arg\max_i V(\{i\})$
        \item Compute $S_G$ greedily:
            \begin{itemize}
                \item repeatedly add element with \emph{highest
                    marginal-value-per-cost}:
                    \begin{align*}
                        i = \argmax_{j\in[N]\setminus
                        S}\frac{V(S\cup\{i\}) - V(S)}{c_i}\label{greedy}
                    \end{align*}
                \item stop when the budget is exhausted
            \end{itemize}
        \item Return:
		\begin{itemize}
		\item $i^*$ if $V(\{i^*\})>V(S_G)$
		\item $S_G$ otherwise
		\end{itemize}
	\end{itemize}
    \end{block}

\vspace*{2em}
\pause

Two problems:
\begin{itemize}
\item $\max$ breaks monotonicity
    \pause
\item threshold payments exceed budget
\end{itemize}
\end{frame}

\begin{frame}{Randomized  Mechanism for Submodular $V$}
    \begin{block}{Allocation rule}
        \begin{itemize}
        \item Compute $i^* = \arg\max_i V(\{i\})$
        \item Compute $S_G$ greedily:
            \begin{itemize}
                \item repeatedly add element with \emph{highest
                    marginal-value-per-cost}:
                    \begin{align*}
                        i = \argmax_{j\in[N]\setminus
                        S}\frac{V(S\cup\{i\}) - V(S)}{c_i}\label{greedy}
                    \end{align*}
                \item \alt<1>{stop when the budget is exhausted}{\alert<2>{stop when $c_k\leq\frac{B}{2}\frac{V(S_G\cup\{i\}) - V(S_G)}{V(S_G\cup\{i\})}$}}
            \end{itemize}
        \item Return:
            \begin{itemize}
                    \alt<1-2>{
		\item $i^*$ if $V(\{i^*\})>V(S_G)$
    \item $S_G$ otherwise}{
        \item
\alert<3>{Select at random between $i^*$ and $S_G$}}
\end{itemize}
    \end{itemize}
\end{block}
\vspace{0.5cm}
\begin{overprint}
\onslide<4>
[Singer 2010] Along with threshold payments, this mechanism is:
\begin{itemize}
\item poly-time,
\item universally truthful, 
\item budget-feasible
\item 7.91-approximate [Chen \etal\ 2011].
\end{itemize}
\onslide<5->
 \vspace{0.5cm}
    \alert{Problem:} $OPT\leq 7.91 \cdot \alert{\mathbb{E}[V(S)]}$\ldots
\end{overprint}
\end{frame}

\begin{frame}{Deterministic Mechanism for Submodular $V$}
    \alert{Idea:} Use a proxy to compute the maximum
    \begin{block}{Allocation rule}
        \begin{itemize}
        \item Compute $i^* = \arg\max_i V(\{i\})$
        \item Compute $S_G$ greedily
        \item Return:
            \begin{itemize}
                \item $i^*$ if $V(\{i^*\}) \geq \alt<1>{V(S_G)}{\alert<2>{OPT_{-i^*}}}$ 
                \item $S_G$ otherwise
            \end{itemize}
        \end{itemize}
    \end{block}
    \vspace{0.5cm}
\uncover<3->{
[Singer 2010] Along with threshold payments, this mechanism is:
\begin{itemize}
\item truthful, 
\item budget-feasible
\item 8.34-approximate [Chen \etal\ 2011].
\end{itemize}
}   
\uncover<4->
{ \vspace{0.5cm}
    \alert{Problem:} $OPT_{-i^*}$ is NP-hard to compute\ldots
}
\end{frame}


\begin{frame}{Blueprint for Deterministic, Poly-time Algorithm}
Azar and Gamzu 2008, Singer 2010, Chen \etal\ 2011:
    \begin{block}{Allocation rule}
        \begin{itemize}
        \item Find $i^* = \arg\max_i V(\{i\})$
        \item Compute $S_G$ greedily
        \item Return:
            \begin{itemize}
                \item $\{i^*\}$ if $V(\{i^*\}) \geq \alert{L^*}$ 
                \item $S_G$ otherwise
            \end{itemize}
        \end{itemize}
    \end{block}
    \vspace{0.5cm}
    \begin{columns}[c]
        \begin{column}{0.53\textwidth}
Replace $OPT$ with a \alert{relaxation $L^*$}:\pause
            \begin{itemize}
		   \item computable in polynomial time
                    \pause
                \item close to $OPT_{-i^*}$
		    \pause
		\item monotone in costs $c$
            \end{itemize}
 	\end{column}
	\pause
        \begin{column}{0.45\textwidth}
   	\begin{itemize}
                \item \textsc{Knapsack} (Chen \etal, 2011)
                \item \textsc{Coverage} (Singer, 2012)
            \end{itemize}
	\end{column}
       \end{columns}
\end{frame}


\begin{frame}{Which relaxation?}
\alt<1>{\begin{block}{Submodular Maximization}
\vspace*{-0.4cm}		
\begin{align*}
			\text{Maximize}\qquad &V(S) \\
			\text{subj. to}\qquad &\textstyle\sum_{i\in S}c_i\leq B
\end{align*}
\end{block}
}
{
\begin{block}{Multilinear Relaxation}
\vspace*{-0.6cm}		
\begin{align*}
    \text{Maximize}\qquad &\alert<2>{F(\lambda)=\mathbb{E}_{S\sim\lambda}[V(S)]}\\
			\text{subj. to}\qquad &\textstyle\sum_{i\in [N]}\lambda_i c_i\leq B,\\& \lambda_i\in[0,1], i\in [N]
\end{align*}
\vspace*{-0.6cm}		
\end{block}
}
\medskip
\pause
For $\lambda^*$ the optimal solution:

\begin{itemize}
    \item
        $V(S) = F(\lambda)\text{ for }\lambda_i=\mathbbm{1}_{i\in S}\quad\visible<3->{\Rightarrow\quad OPT \leq F(\lambda^*).}$ 
        \pause
    \item $F(\lambda^*)$ is monotone in $c$
        \pause
    \item Pipage rounding [Ageev and Sviridenko]:
        \begin{displaymath}F(\lambda^*) \leq OPT + V(i^*)\end{displaymath}
\end{itemize}
\vspace{1cm}
\pause
Good relaxation candidate…\pause{} if it can be computed in poly-time.
\end{frame}

\begin{frame}{A relaxation for \textsc{EDP}}
\alt<1>{
    \begin{block}{\textsc{EDP} }
        \vspace*{-0.4cm}		
        \begin{align*}
            \text{Maximize}\qquad &V(S) = \log \det\big(I+\sum_{i\in S}x_ix_i^T\big) \\
            \text{subj. to}\qquad &\textstyle\sum_{i\in S}c_i\leq B
        \end{align*}
    \end{block}
}
{
    \begin{block}{Convex Relaxation}
        \vspace*{-0.4cm}		
        \begin{align*}
            \text{Maximize}\qquad &\alert<2>{L(\lambda)} = \log \det \big(I+\sum_{i\in [N]}\alert<2>{\lambda_i}x_ix_i^T\big) \\
            \text{subj. to}\qquad &\textstyle\sum_{i\in [N]}\alert<2>{\lambda_i}c_i\leq B, \lambda_i\in [0,1]
        \end{align*}
    \end{block}
}
\pause
\vspace{0.5cm}
$L(\lambda^*)$ is:
\begin{itemize}
    \item Monotone in $c$
        \pause
    \item Poly-time: $L$ is concave
        \pause
    \item Close to $OPT$:\pause
        \vspace{0.5cm}
        \begin{block}{Technical Lemma}
            \begin{displaymath}
                OPT\leq  L(\lambda^*) \leq 2 OPT + 2V(\{i^*\})
            \end{displaymath}
        \end{block}
\end{itemize}
\end{frame}

\begin{frame}{Proof Steps}
           \begin{block}{Technical Lemma}
                \begin{displaymath}
                 OPT\leq   L(\lambda^*) \leq 2 OPT + 2V(\{i^*\})
                \end{displaymath}
            \end{block}
            \uncover<2->{Relate $L$ to the multi-linear extension $F$:
            \begin{itemize}
                \item $F(\lambda)=\mathbb{E}_{S\sim\lambda}\Big[\log\det\big(I+\sum_{i\in S}x_ix_i^T\big)\Big]$
                \item $L(\lambda)=\log\det\Big(I+\mathbb{E}_{S\sim\lambda}\big[\sum_{i\in S}x_ix_i^T\big]\Big)$
            \end{itemize}
\bigskip}
\uncover<3->{
    Bounding the derivatives of $\log\det$ $\Rightarrow$ $L(\lambda)\leq 2 F(\lambda)$
}
\bigskip

\uncover<4>{Finally, $F(\lambda^*)\leq OPT+V(\{i^*\})$ (pipage rounding).}
\end{frame}

\begin{frame}{Summary}
    \begin{block}{Allocation rule}
        \begin{itemize}
        \item Compute $i^* = \arg\max_i V(\{i\})$
        \item Compute $S_G$ greedily
        \item Compute $L(\lambda^*) = \max_{\lambda}\Big\{\log \det \big(I+\sum_{i\in [N]}\lambda_ix_ix_i^T\big),\; \sum_{i\in[N]}\lambda_ic_i\leq B\Big\}$
        \item Return:
            \begin{itemize}
                \item $i^*$ if $V(\{i^*\}) \geq L(\lambda^*)$ 
                \item $S_G$ otherwise
            \end{itemize}
        \end{itemize}
    \end{block}
    \vspace{0.5cm}
    \pause
    Along with threshold payments:
    \begin{itemize}
        \item individually rational, truthful, budget-feasible
            \pause
        \item poly-time
            \pause
        \item 12.98-approximate
    \end{itemize}
\end{frame}

\begin{frame}{}
    \begin{center}
    \Huge{\alert{Are we done?}}
    \end{center}
\end{frame}

\begin{frame}{Tiny glitch}
    \begin{block}{Allocation rule}
        \begin{itemize}
            \item Compute $i^* = \arg\max_i V(\{i\})$
            \item Compute $S_G$ greedily
            \item Compute $\alert<2>{L(\lambda^*) = \max_{\lambda}\Big\{\log \det \big(I+\sum_{i\in [N]}\lambda_ix_ix_i^T\big),\; \sum_{i\in[N]}\lambda_ic_i\leq B\Big\}}$
            \item Return:
                \begin{itemize}
                    \item $i^*$ if $V(\{i^*\}) \geq L(\lambda^*)$ 
                    \item $S_G$ otherwise
                \end{itemize}
        \end{itemize}
    \end{block}
    \pause
    \vspace{0.5cm}
    \begin{itemize}
        \item $L(\lambda^*)$ is monotone in $c$
            \pause
        \item but not an approximation of $L(\lambda^*)$
    \end{itemize}
\end{frame}

\begin{frame}{$\varepsilon$-truthfulness}
    Assume that:
    \begin{displaymath}
        c_i'\leq c_i-\varepsilon \Rightarrow L(\lambda^*_{c'})\geq L(\lambda^*_c)+\delta
    \end{displaymath}

    \pause
    \vspace{1cm}
    Then, if $\tilde{L}(\lambda^*_c)$ is a $\delta$-approximate of $L(\lambda^*_c)$
    \begin{displaymath}
        c_i'\leq c_i-\varepsilon \Rightarrow \tilde{L}(\lambda^*_{c'})\geq \tilde{L}(\lambda^*_c)
    \end{displaymath}

    \vspace{1cm}
    \pause
    Implies $\varepsilon$-truthfulness:
    \begin{center}
        \alert{users have not incentive to lie by more $\varepsilon$}
    \end{center}
\end{frame}

\begin{frame}{Sensitivity analysis}
    \begin{block}{Goal ($\varepsilon$, $\delta$)-monotonicity}
        \begin{displaymath}
            c_i'\leq c_i-\varepsilon \Rightarrow L(\lambda^*_{c'})\geq L(\lambda^*_c)+\delta
        \end{displaymath}
    \end{block}
    \pause
    \vspace{0.5cm}

    Bad news:
    \begin{displaymath}
        \frac{\partial L(\lambda^*_c)}{\partial c_i} \propto \lambda_i^*\only<3->{\alert{\geq \alpha}}
    \end{displaymath}

    \pause
    \vspace{0.5cm}
    Solution: regularize the problem
    \begin{displaymath}
        L(\lambda^*) = \max_{\lambda}\Big\{\log \det \big(I+\sum_{i\in [N]}\lambda_ix_ix_i^T\big),\; \sum_{i\in[N]}\lambda_ic_i\leq B,\; \alert{\lambda_i\geq \alpha}\Big\}
    \end{displaymath}

    \pause
    \vspace{0.5cm}
    Then carefully tune all the parameters…
\end{frame}


\section{Generalization}

\begin{frame}{Outline}
    \tableofcontents[currentsection]
\end{frame}

\begin{frame}{Towards More General ML Tasks}
   \begin{center}
        \includegraphics<1>[scale=0.4]{st2.pdf}
        \includegraphics<2>[scale=0.4]{st3.pdf}
        \includegraphics<3>[scale=0.4]{st4.pdf}
        \includegraphics<4-5>[scale=0.4]{st25.pdf}
        \includegraphics<6->[scale=0.4]{st26.pdf}
    \end{center}

    \begin{overprint}
        \onslide<1>
		\begin{center}$N$ experiment subjects\end{center}
       	\onslide<2>
            \begin{center}
                $x_i\in \Omega$: public features
            \end{center}
        \onslide<3>
            \begin{center}
                $y_i\in \reals$: private data 
            \end{center}
        \onslide<4-5>
             \begin{center}
                \textbf{Hypothesis.}  There exists $h:\Omega\to \reals$ s.t.
 %   \begin{displaymath}
        $y_i =  h(x_i) + \varepsilon_i,$ %\quad
	where $\varepsilon_i$ are independent.
	\visible<5>{Examples: Logistic Regression, LDA, SVMs, \etc}
  % \end{displaymath}
		\end{center}
%$$y_i = \beta^Tx_i + \varepsilon_i, i=1,\ldots, N$$ 
            %\end{center}
	    \onslide<6>
	    \begin{center}
		Experimenter {\tt E} wishes to learn $h$, and has a prior distribution over $$\mathcal{H}=\{\text{mappings from }\Omega\text{ to }\reals\}.$$ 
	    \end{center}
    \onslide<7->\begin{center}
\vspace*{-0.5cm}
%    \begin{block}
%{Value function: Information gain}
        \begin{displaymath}
            V(S) = H(h) - H(h\mid y_i,i\in S),\quad S\subseteq[N]
        \end{displaymath}
 %   \end{block}
	\visible<8>{$V$ is submodular! [Krause and Guestrin 2009]}
    \end{center}
	\end{overprint}

\end{frame}

\begin{comment}
    \begin{itemize}
        \item Generative model: $y_i = f(x_i) + \varepsilon_i,\;i\in\mathcal{A}$
        \pause
        \item prior knowledge of the experimenter: $f$ is a \alert{random variable}
        \pause
        \item \alert{uncertainty} of the experimenter: entropy $H(f)$
        \pause
        \item after observing $\{y_i,\; i\in S\}$, uncertainty: $H(f\mid S)$
    \end{itemize}

    \pause
    \vspace{0.5cm}

    \begin{block}{Value function: Information gain}
        \begin{displaymath}
            V(S) = H(f) - H(f\mid S),\quad S\subset\mathcal{A}
        \end{displaymath}
    \end{block}

    \pause
    \vspace{0.5cm}

    $V$ is \alert{submodular} $\Rightarrow$ randomized budget feasible mechanism
\end{frame}

\end{comment}

\begin{frame}{Conclusion}
    \begin{itemize}
        \item Experimental design + Budget Feasible Mechanisms
        \vspace{1cm}
        \pause
        \item Deterministic mechanism for other ML Tasks? General submodular functions?
            \vspace{1cm}
            \pause
        \item  Approximation ratio $\simeq \alert{13}$. Lower bound: \alert{$2$}
    \end{itemize}
\end{frame}
\begin{frame}{}
\vspace{\stretch{1}}
\begin{center}
{\Huge Thank you!}
\end{center}
\vspace{\stretch{1}}
\end{frame}

\begin{frame}{Non-Homotropic Prior}
Recall that:
\begin{align*}
V(S)&=H(\beta)-H(\beta\mid y_i,i\in S)\\
& = \frac{1}{2}\log\det (R^{-1}+\sum_{i\in S}x_ix_i^T)
\end{align*}
What if $R\neq I$?
\begin{theorem}
 There exists a truthful, poly-time, and budget feasible mechanism for the objective
 function $V$ above. Furthermore, 
 the algorithm computes a set $S^*$ such that:
 \begin{displaymath}
    OPT \leq 
    \frac{5e-1}{e-1}\frac{2}{\mu\log(1+1/\mu)}V(S^*) + 5.1 
\end{displaymath}
where $\mu$ is the largest eigenvalue of $R$.
\end{theorem}
\end{frame}

\end{document}